Page 213 - Servo Motors and Industrial Control Theory

Appendix C 211

current limiter is built to protect the DC motor from initial large current. The fol-
lowing data represents the property of the separately excited DC motor selected
for this application and the mass of the table;
Mass of the moving table = 50 kg
Motor power rating = 1 kW
Rotor inertia = 0.0093 kg m 2
Rated motor velocity = 1260 rpm
Maximum torque limit = 113 N m
Armature resistance = 0.58 Ω
Armature inductance = 0.0023 H
Torque constant = 0.83 (Nm/amp)
Voltage constant = 0.83 (volts/(rad/sec))
Write the governing differential equations for various parts of the system assum-
ing that because of high accuracy requirements the inductance and lead screw
stiffness cannot be ignored. Assume that the static and viscous friction in the
motor and lead screw is negligible. For a complicated system as it is the case
for this application, it is better to write the equations in state space form without
reducing the differential equations. First assume that a proportional controller
must be used. Calculate the proper values of the proportional control and the
derivative gains to give a maximum speed of response with a damping ratio
of at least 0.7 in the dominant eigenvalues of the system. Hence determine the
dynamic response time of the system for small variation of the demand posi-
tion. Remember that this only gives the dynamic response time and to find the
total response time for large demand signal you must consider the case when the
motor generates maximum torque. In this case assuming that the motion starts
from rest simply you should solve equation of motion derived from the Newton
second law of motion. For this exercise determine the total response time when
the table has to be moved 50 cm from rest. Your state space model must have
two input variables of demand signal and the force applied to the motor. You
should note that the velocity is difficult to measure from the table. In a practical
situation a small DC motor might be connected at the end of lead screw to give
the velocity of the table.
Find the steady state error for unit step input of demand signal which in this case
is one meter because the metric system of units has to be used. It is not realistic
but it gives some idea about the steady state error involved in the system. Also
determine the steady state error when step input of 100 N force is applied to the
motor.
Suppose that the speed of response and the steady state errors are not satisfac-
tory for this application. In order to improve the performance it is proposed that
state variable feedback control strategy must be used. Add an integrator in the
forward loop to ensure zero steady state error. The order of the state equation
increases to six and you should define six state variables that are measurable for
direct feedback. For this you should write the governing differential equation in
an intelligent form so that all state variables are measurable for direct feedback.